
Using the fact that two maps can be locally embedded in a single map, it is shown in [1]} that these sheaves glue together into a single sheaf \(K_{X,s}\)  on \(X^{red}\)  called the canonical sheaf, with natural local isomorphisms:
\(&\iota _{R,U,f,i}:K_{X,s}|_{R^{red}}\rightarrow i^\ast (K_U^{\otimes ^2})|_{R^{red}}\nonumber \\&J_\Phi \circ \iota _{R,U,f,i}=\iota _{S,V,g,j}\) 
